Not a Luna user, but if I understand your question I think that you are referring to the prompt string.
The terminal state when opened might indeed depend on how it was opened, in this case the directory where it was opened.
~$ indicates that it was opened to your HOME directory.
~/Desktop/DEBIAN$ indicates that it was opened to the Desktop/DEBIAN/ directory below your HOME directory (~/ plus Desktop/DEBIAN).
So the "apps" method must include a default path whereas the ctl-alt-t method defaults to your HOME directory.
It makes no other difference. If you open to Desktop/DEBIAN but want to be in your HOME, simply type cd <ENTER> and you will be magically transported there!
If you are not familiar with navigation and other terminal topics, a good place to start is Bash Beginners Guide
or Advanced Bash Scripting Guide
You can also find how to change the contents of that prompt string while there!
If using some other shell see the respective man page.
Hope this helps.